Regarding the corresponding threshold in random graphs, Montgomery [1]} proved that w.h.p. for any given spanning tree with degrees bounded by \(\Delta \) , the random graph \(\mathbb {G}(n,O_\Delta (\log n/n))\)  contains a copy. (In fact, Montgomery proved the stronger universality result that w.h.p. \(\mathbb {G}(n,O_\Delta (\log n /n))\)  contains a copy of every spanning tree with maximal degree at most \(\Delta \) .) Note that a lower bound of \(\log n /n\)  for this threshold follows from considering the presence of isolated vertices. Therefore Montgomery's result is tight up to a multiplicative constant.
